A combinatorial ${E_\infty}$-algebra structure on cubical cochains and the Cartan-Serre map

TL;DR

This paper constructs an explicit combinatorial ${E_}$-algebra structure on cubical cochains, extending their product, and proves the Cartan-Serre map preserves this rich algebraic structure as a quasi-isomorphism.

Contribution

It introduces a full ${E_}$-structure on cubical cochains and demonstrates the Cartan-Serre map is an ${E_}$-quasi-isomorphism, advancing algebraic topology methods.

Findings

Constructed an explicit ${E_}$-structure on cubical cochains.

Proved the Cartan-Serre map is an ${E_}$-quasi-isomorphism.

Extended the algebraic understanding of cubical cochains.

Abstract

Cubical cochains are equipped with an associative product, dual to the Serre diagonal, lifting the graded commutative structure in cohomology. In this work we introduce through explicit combinatorial methods an extension of this product to a full ${E_\infty}$-structure. As an application we prove that the Cartan-Serre map, which relates the cubical and simplicial singular cochains of spaces, is a quasi-isomorphism of ${E_\infty}$-algebras.